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Every finite graph arises as the singular set of a compact 3‐D calibrated area minimizing surface Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2024-03-09 Zhenhua Liu
Given any (not necessarily connected) combinatorial finite graph and any compact smooth 6‐manifold with the third Betti number , we construct a calibrated 3‐dimensional homologically area minimizing surface on equipped in a smooth metric , so that the singular set of the surface is precisely an embedding of this finite graph. Moreover, the calibration form near the singular set is a smoothly twisted
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Delta‐convex structure of the singular set of distance functions Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2024-03-07 Tatsuya Miura, Minoru Tanaka
For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta‐convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta‐convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean
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Non‐degenerate minimal submanifolds as energy concentration sets: A variational approach Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2024-02-28 Guido De Philippis, Alessandro Pigati
We prove that every non‐degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg–Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet–Jerrard–Sternberg. The same proof applies also to the ‐Yang–Mills–Higgs
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A Liouville‐type theorem for cylindrical cones Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2024-02-23 Nick Edelen, Gábor Székelyhidi
Suppose that is a smooth strictly minimizing and strictly stable minimal hypercone (such as the Simons cone), , and a complete embedded minimal hypersurface of lying to one side of . If the density at infinity of is less than twice the density of , then we show that , where is the Hardt–Simon foliation of . This extends a result of L. Simon, where an additional smallness assumption is required for
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Diameter estimates in Kähler geometry Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2024-02-22 Bin Guo, Duong H. Phong, Jian Song, Jacob Sturm
Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for estimates for the Monge–Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, we solve the long‐standing problem of uniform diameter bounds
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Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-12-21 Ujan Gangopadhyay, Subhroshekhar Ghosh, Kin Aun Tan
Gibbsian structure in random point fields has been a classical tool for studying their spatial properties. However, exact Gibbs property is available only in a relatively limited class of models, and it does not adequately address many random fields with a strongly dependent spatial structure. In this work, we provide a very general framework for approximate Gibbsian structure for strongly correlated
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Arnold diffusion in Hamiltonian systems on infinite lattices Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-12-17 Filippo Giuliani, Marcel Guardia
We consider a system of infinitely many penduli on an m-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying
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Chord measures in integral geometry and their Minkowski problems Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-12-11 Erwin Lutwak, Dongmeng Xi, Deane Yang, Gaoyong Zhang
To the families of geometric measures of convex bodies (the area measures of Aleksandrov-Fenchel-Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their
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Overcrowding and separation estimates for the Coulomb gas Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-12-04 Eric Thoma
We prove several results for the Coulomb gas in any dimension d ≥ 2 $d \ge 2$ that follow from isotropic averaging, a transport method based on Newton's theorem. First, we prove a high-density Jancovici–Lebowitz–Manificat law, extending the microscopic density bounds of Armstrong and Serfaty and establishing strictly sub-Gaussian tails for charge excess in dimension 2. The existence of microscopic
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The anisotropic min-max theory: Existence of anisotropic minimal and CMC surfaces Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-12-01 Guido De Philippis, Antonio De Rosa
We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension
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Critical sets of solutions of elliptic equations in periodic homogenization Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-11-20 Fanghua Lin, Zhongwei Shen
In this paper we study critical sets of solutions u ε $u_\varepsilon$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first-order correctors, we show that the ( d − 2 ) $(d-2)$ -dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε, provided that doubling indices for
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Infinite order phase transition in the slow bond TASEP Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-11-20 Sourav Sarkar, Allan Sly, Lingfu Zhang
In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to 1−ε$1-\varepsilon$ for some small ε>0$\varepsilon >0$. Janowsky and Lebowitz posed the well-known question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations
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Subquadratic harmonic functions on Calabi-Yau manifolds with maximal volume growth Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-11-16 Shih-Kai Chiu
On a complete Calabi-Yau manifold M $M$ with maximal volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving a Liouville-type theorem for harmonic 1-forms, which follows from a new local L 2 $L^2$ estimate of the exterior derivative.
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Global solutions of the compressible Euler-Poisson equations with large initial data of spherical symmetry Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-11-13 Gui-Qiang G. Chen, Lin He, Yong Wang, Difan Yuan
We are concerned with a global existence theory for finite-energy solutions of the multidimensional Euler-Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main challenges is the strengthening of waves as they move radially inward towards the origin, especially under the self-consistent gravitational field for gaseous stars.
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Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-11-13 William M. Feldman
We apply new results on free boundary regularity to obtain a quantitative convergence rate for the shape optimizers of the first Dirichlet eigenvalue in periodic homogenization. We obtain a linear (with logarithmic factors) convergence rate for the optimizing eigenvalue. Large scale Lipschitz free boundary regularity of almost minimizers is used to apply the optimal L2$L^2$ homogenization theory in
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A dynamical approach to the study of instability near Couette flow Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-11-08 Hui Li, Nader Masmoudi, Weiren Zhao
In this paper, we obtain the optimal instability threshold of the Couette flow for Navier–Stokes equations with small viscosity ν > 0 $\nu >0$ , when the perturbations are in the critical spaces H x 1 L y 2 $H^1_xL_y^2$ . More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size ν 1 2 − δ 0 $\nu ^{\frac{1}{2}-\delta _0}$ with any small δ 0 > 0 $\delta
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The maximum of log-correlated Gaussian fields in random environment Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-11-02 Florian Schweiger, Ofer Zeitouni
We study the distribution of the maximum of a large class of Gaussian fields indexed by a box VN⊂Zd$V_N\subset \mathbb {Z}^d$ and possessing logarithmic correlations up to local defects that are sufficiently rare. Under appropriate assumptions that generalize those in Ding et al., we show that asymptotically, the centered maximum of the field has a randomly-shifted Gumbel distribution. We prove that
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Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-30 Siva Athreya, Oleg Butkovsky, Khoa Lê, Leonid Mytnik
We study stochastic reaction–diffusion equation
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Integrability of SLE via conformal welding of random surfaces Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-19 Morris Ang, Nina Holden, Xin Sun
We demonstrate how to obtain integrability results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called SLEκ(ρ−;ρ+)$\operatorname{SLE}_\kappa (\rho _-;\rho _+)$. Our proof is built
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On the incompressible limit for a tumour growth model incorporating convective effects Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-16 Noemi David, Markus Schmidtchen
In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between
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Log-Sobolev inequality for the φ24 and φ34 measures Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-16 Roland Bauerschmidt, Benoit Dagallier
The continuum φ24$\varphi ^4_2$ and φ34$\varphi ^4_3$ measures are shown to satisfy a log-Sobolev inequality uniformly in the lattice regularisation under the optimal assumption that their susceptibility is bounded. In particular, this applies to all coupling constants in any finite volume, and uniformly in the volume in the entire high temperature phases of the φ24$\varphi ^4_2$ and φ34$\varphi ^4_3$ models
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Log-Sobolev inequality for near critical Ising models Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-16 Roland Bauerschmidt, Benoit Dagallier
For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log-Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the log-Sobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover
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Magnetic helicity, weak solutions and relaxation of ideal MHD Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-08 Daniel Faraco, Sauli Lindberg, László Székelyhidi
We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics (MHD) in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations
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Soft Riemann-Hilbert problems and planar orthogonal polynomials Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-08 Haakan Hedenmalm
Riemann-Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, for example, in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix-valued Riemann-Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest
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Local laws and a mesoscopic CLT for β-ensembles Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-08 Luke Peilen
We study the statistical mechanics of the log-gas, or β-ensemble, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on the next order energy that are valid down to microscopic length scales. To our knowledge, this is the first time that this kind of a local quantity has been controlled for the log-gas. Simultaneously, we exhibit a control on fluctuations
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Hölder regularity of the Boltzmann equation past an obstacle Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-06 Chanwoo Kim, Donghyun Lee
Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in Cx,v0,12−$C^{0,\frac{1}{2}-}_{x,v}$ for the Boltzmann equation of the hard-sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle
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Conformal covariance of connection probabilities and fields in 2D critical percolation Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-05 Federico Camia
Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well-defined scaling limit for every n ≥ 2 $n \ge 2$ . Moreover, the limiting functions P n ( x 1 , … , x n
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Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-05 Costante Bellettini
We address the one-parameter minmax construction for the Allen–Cahn energy that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold Nn+1$N^{n+1}$ with n≥2$n\ge 2$ (Guaraco's work, relying on works by Hutchinson, Tonegawa, and Wickramasekera when sending the Allen–Cahn parameter to 0). We obtain the following result: if the Ricci
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Sharp asymptotic estimates for expectations, probabilities, and mean first passage times in stochastic systems with small noise Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-05 Tobias Grafke, Tobias Schäfer, Eric Vanden-Eijnden
Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived
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Stationary measure for the open KPZ equation Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-05 Ivan Corwin, Alisa Knizel
We provide the first construction of stationary measures for the open KPZ equation on the spatial interval [0,1] with general inhomogeneous Neumann boundary conditions at 0 and 1 depending on real parameters u and v, respectively. When u+v≥0$u+v\ge 0$, we uniquely characterize the constructed stationary measures through their multipoint Laplace transform, which we prove is given in terms of a stochastic
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High-dimensional limit theorems for SGD: Effective dynamics and critical scaling Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-04 Gérard Ben Arous, Reza Gheissari, Aukosh Jagannath
We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic
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Sine-kernel determinant on two large intervals Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-04 Benjamin Fahs, Igor Krasovsky
We consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.
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Almost monotonicity formula for H-minimal Legendrian surfaces in the Heisenberg group Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-03 Tristan Rivière
We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called contact stationary Legendrian immersions or Hamiltonian stationary immersions) in the Heisenberg Group H2${\mathbb {H}}^2$. From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. We also present some possible range of applications of this formula.
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Directed mean curvature flow in noisy environment Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-03 Andris Gerasimovičs, Martin Hairer, Konstantin Matetski
We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity
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The forbidden region for random zeros: Appearance of quadrature domains Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-10-02 Alon Nishry, Aron Wennman
Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zeros of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero
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Optimal regularity for supercritical parabolic obstacle problems Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-29 Xavier Ros-Oton, Clara Torres-Latorre
We study the obstacle problem for parabolic operators of the type ∂t+L$\partial _t + L$, where L is an elliptic integro-differential operator of order 2s, such as (−Δ)s$(-\Delta )^s$, in the supercritical regime s∈(0,12)$s \in (0,\frac{1}{2})$. The best result in this context was due to Caffarelli and Figalli, who established the Cx1,s$C^{1,s}_x$ regularity of solutions for the case L=(−Δ)s$L = (-\Delta
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Spectrum of random d-regular graphs up to the edge Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-28 Jiaoyang Huang, Horng-Tzer Yau
Consider the normalized adjacency matrices of random d-regular graphs on N vertices with fixed degree d⩾3$d\geqslant 3$. We prove that, with probability 1−N−1+ε$1-N^{-1+\varepsilon }$ for any ε>0$\varepsilon >0$, the following two properties hold as N→∞$N \rightarrow \infty$ provided that d⩾3$d\geqslant 3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay
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Erratum for “Global Identifiability of Differential Models” Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-22 Hoon Hong, Alexey Ovchinnikov, Gleb Pogudin, Chee Yap
We are grateful to Peter Thompson for pointing out an error in [1, Lemma 3.5, p. 1848]. The original proof worked only under the assumption that θ̂$\hat{\theta }$ is a vector of constants. However, some of the components of θ̂$\hat{\bm{\theta }}$ could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with θ̂$\hat{\bm{\theta }}$ involving states) later
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Discrete honeycombs, rational edges, and edge states Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-22 Charles L. Fefferman, Sonia Fliss, Michael I. Weinstein
Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type”, generalizing the classical zigzag and armchair edges. We prove that zero energy / flat band edge states arise for edges of zigzag type, but never for those
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An upper Minkowski dimension estimate for the interior singular set of area minimizing currents Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-18 Anna Skorobogatova
We show that for an area minimizing m-dimensional integral current T of codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most m − 2 $m-2$ . This provides a strengthening of the existing ( m − 2 ) $(m-2)$ -dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the proof
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Logarithmic cotangent bundles, Chern-Mather classes, and the Huh-Sturmfels involution conjecture Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-15 Laurenţiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu
Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu-Zhou. The first application of our formula is a geometric description of Chern-Mather classes of an arbitrary very affine variety, generalizing earlier
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Landscape complexity beyond invariance and the elastic manifold Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-14 Gérard Ben Arous, Paul Bourgade, Benjamin McKenna
This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self-interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as
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Phase diagram and topological expansion in the complex quartic random matrix model Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-14 Pavel Bleher, Roozbeh Gharakhloo, Kenneth T-R McLaughlin
We use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers N j ( g ) $\mathcal {N}_j(g)$ of 4-valent connected graphs with j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann
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Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-14 José A. Carrillo, Ruiwen Shu
We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations
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Complex analytic dependence on the dielectric permittivity in ENZ materials: The photonic doping example Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-14 Robert V. Kohn, Raghavendra Venkatraman
Motivated by the physics literature on “photonic doping” of scatterers made from “epsilon-near-zero” (ENZ) materials, we consider how the scattering of time-harmonic TM electromagnetic waves by a cylindrical ENZ region Ω × R $\Omega \times \mathbb {R}$ is affected by the presence of a “dopant” D ⊂ Ω $D \subset \Omega$ in which the dielectric permittivity is not near zero. Mathematically, this reduces
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Magnetic slowdown of topological edge states Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-12 Guillaume Bal, Simon Becker, Alexis Drouot
We study the propagation of wavepackets along curved interfaces between topological, magnetic materials. Our Hamiltonian is a massive Dirac operator with a magnetic potential. We construct semiclassical wavepackets propagating along the curved interface as adiabatic modulations of straight edge states under constant magnetic fields. While in the magnetic-free case, the wavepackets propagate coherently
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Thermodynamic limit of the first Lee-Yang zero Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-11 Jianping Jiang, Charles M. Newman
We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in R${\mathbb {R}}$ of finite-volume singularities in C${\mathbb {C}}$. For the Ising model defined on a finite Λ⊂Zd$\Lambda \subset \mathbb {Z}^d$ at inverse temperature β≥0$\beta \ge 0$ and external field h, let α1(Λ,β)$\alpha _1(\Lambda ,\beta )$ be the modulus of the first zero
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Compressive phase retrieval: Optimal sample complexity with deep generative priors Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-11 Paul Hand, Oscar Leong, Vladislav Voroninski
Advances in compressive sensing (CS) provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able
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A scaling limit of the parabolic Anderson model with exclusion interaction Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-10 Dirk Erhard, Martin Hairer
We consider the (discrete) parabolic Anderson model ∂ u ( t , x ) / ∂ t = Δ u ( t , x ) + ξ t ( x ) u ( t , x ) $\partial u(t,x)/\partial t=\Delta u(t,x) +\xi _t(x) u(t,x)$ , t ≥ 0 $t\ge 0$ , x ∈ Z d $x\in \mathbb {Z}^d$ , where the ξ-field is R $\mathbb {R}$ -valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly
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Fermi isospectrality for discrete periodic Schrödinger operators Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-10 Wencai Liu
Let Γ = q 1 Z ⊕ q 2 Z ⊕ … ⊕ q d Z $\Gamma =q_1\mathbb {Z}\oplus q_2 \mathbb {Z}\oplus \ldots \oplus q_d\mathbb {Z}$ , where q l ∈ Z + $q_l\in \mathbb {Z}_+$ , l = 1 , 2 , … , d $l=1,2,\ldots ,d$ , are pairwise coprime. Let Δ + V $\Delta +V$ be the discrete Schrödinger operator, where Δ is the discrete Laplacian on Z d $\mathbb {Z}^d$ and the potential V : Z d → C $V:\mathbb {Z}^d\rightarrow \mathbb
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Pure gravity traveling quasi-periodic water waves with constant vorticity Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-09 Massimiliano Berti, Luca Franzoi, Alberto Maspero
We prove the existence of small amplitude time quasi-periodic solutions of the pure gravity water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space periodic free interface. Using a Nash-Moser implicit function iterative scheme we construct traveling nonlinear waves which pass through each other slightly deforming and retaining forever a quasiperiodic
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Critical local well-posedness for the fully nonlinear Peskin problem Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-08 Stephen Cameron, Robert M. Strain
We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for
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Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the (n−2)$(n-2)$-area functional Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-07 Davide Parise, Alessandro Pigati, Daniel Stern
Given a hermitian line bundle L → M $L\rightarrow M$ on a closed Riemannian manifold ( M n , g ) $(M^n,g)$ , the self-dual Yang–Mills–Higgs energies are a natural family of functionals
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C2,α$C^{2,\alpha }$ regularity of free boundaries in optimal transportation Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-07 Shibing Chen, Jiakun Liu, Xu-Jia Wang
The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the C 2 , α $C^{2,\alpha }$ regularity of the potential function as well as that of the free boundary, thereby resolve
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Constrained deformations of positive scalar curvature metrics, II Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-07 Alessandro Carlotto, Chao Li
We prove that various spaces of constrained positive scalar curvature metrics on compact three-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean-convex and the minimal case. We then discuss the implications of these results on the topology
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Prescribed curvature measure problem in hyperbolic space Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-07 Fengrui Yang
The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider the prescribed curvature measure problem in the hyperbolic space. We obtain the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures ( k < n ) $(k
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Free boundary partial regularity in the thin obstacle problem Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-05 Federico Franceschini, Joaquim Serra
For the thin obstacle problem in R n $\mathbb {R}^n$ , n ≥ 2 $n\ge 2$ , we prove that at all free boundary points, with the exception of a ( n − 3 ) $(n-3)$ -dimensional set, the solution differs from its blow-up by higher order corrections. This expansion entails a C1, 1-type free boundary regularity result, up to a codimension 3 set.
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Existence of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics Comm. Pure Appl. Math. (IF 3.0) Pub Date : 2023-09-05 Yanjin Wang, Zhouping Xin
We establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity